Mere Addition is equivalent to avoiding the Sadistic Conclusion in all plausible variable-population social orderings
Introduction
Population ethics studies the axiomatic properties of social orderings of welfare vectors with different population sizes. Two well-studied axioms in population ethics hinge on the sign of lifetime utilities. Mere Addition, introduced by Parfit (1984), holds that adding a positive-utility life to any population does not result in a worse population. The Sadistic Conclusion, introduced by Arrhenius (2000), holds that adding negative lives can be strictly preferred to adding positive lives. The literature typically describes the Sadistic Conclusion as ethically worse than denying Mere Addition. Arrhenius (2017) calls the requirement to avoid the Sadistic Conclusion a “less controversial assumption” than Mere Addition (p. 297).
Arrhenius (2017) notes that social orderings that violate Mere Addition “tend to imply” the Sadistic Conclusion (p. 94). Greaves (2017) calls these axioms “related”. Bossert (2017) especially highlights the normative link but less generally than we do. We show the link is stronger than tendency. Under a set of basic plausibility axioms not violated by any social ordering defended in the economics literature, these two conditions are equivalent. In their theorem 5.4, Blackorby et al. (2005) state a related result3 that only applies to social orderings that are same-number generalized utilitarian (e.g., not rank-dependent evaluations, an important recent innovation by Asheim and Zuber, 2014).
Section snippets
Setting and basic axioms
Our notation follows Blackorby et al. (2005). is the integers, is the real numbers, and are the positive and nonnegative real numbers, respectively, and similarly for , , and .
Populations are finite-length vectors of real numbers, where the th position in the vector is the lifetime utility of person . Utilities are normalized so that is a neutral life (as good as a life with no experiences) for person . Following Asheim and Zuber (2014), an index enclosed in square
Mere addition and the Sadistic conclusion
Axiom Mere Addition For all and all , .
Axiom Avoidance of the Sadistic Conclusion For all , all , and all , .
Total Utilitarianism () and Total Prioritarianism ( for increasing, concave such that ) satisfy Mere Addition and avoid the Sadistic Conclusion. Average Utilitarianism (), Average Prioritarianism (), Number-Dampened Utilitarianism (, where , yielding a positive, increasing, concave transformation of population size; abbreviated NDGU),
Result
Axiom Consistent Expansion For all , , and , if then there exists such that .
Consistent Expansion holds that if adding -lives to makes a worse combined population, then adding some further number of -lives makes an even worse population.4 Consistent Expansion is satisfied by Total and Average Utilitarianism and Prioritarianism,
Discussion
Ng’s (1989) classic impossibility theorem shows that no social ordering can:
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satisfy Mere Addition,
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satisfy a normatively unobjectionable principle called Non-Antiegalitarianism, and
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avoid Parfit’s (1984) Repugnant Conclusion.
Because the economics literature has never entertained violating Non-Antiegalitarianism, in practice one must reject Mere Addition or accept Parfit’s Repugnant Conclusion. One implication of our result is that, in the context of the specified axioms, satisfying Mere Addition
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Although this paper received no specific funding, both authors’ research is supported by grant NICHD, United States of America grants K01HD098313 and P2CHD042849. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.